Complex Analysis: Set Theory
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I just had a confusing lecture on the following:
Open Set
Closed set
Open and Connected
Domain
Bounded and Unbounded sets
The lecture was confusing to say the least. Could you please give me a clear explanation of these concepts with examples?
By the Way:
Heisenberg and Schrodinger are in a car and are stopped by police officers. One officer asks Heisenberg if he knows how fast he is going. He answers, "No but I know where I am."
The other asks Schrodinger if he is aware of the dead cat in his trunk. He answers, " I am now."
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Solution Summary
The expert examines the complex analysis set theory.
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The set S in the complex plane define some Euclidean area.
We first define the concept of neighborhood. A neighborhood of a point z is the set that contains all points such that
z is an interior point of the set if there exists such that
An interior point of S is a point which is included in S and all the points around it within an infinitesimally small radius are also members of the set S.
All the interior points of S form a set called the interior of S and is denoted by
Now, if we have a set then we have a complementary set , that is if then and vice versa. If a point belongs to S it cannot belong to SC, and if a point belongs to SC it cannot belong to S.
An exterior point of S is defined as an interior point of SC.
That is:
iff
in other words, is an exterior point of S if an only if - there are no points in the neighborhood of z that belong to S.
A boundary point of S is a point which is not interior nor exterior to S, that is, in its neighborhood we find points that belong to S and points that belong to
The set of all such points is called the ...
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